Microloans, a lifeline usually associated with the world's poorest countries, are growing across Europe, giving tens of thousands of people like Fátima Fernández a chance to start small businesses and avoid falling into poverty.

Ms. Fernández, a 31-year-old dance teacher who had been out of work nearly a year, stopped looking for a job and opened a yoga and dance school in Madrid in December 2012 with a €25,000 ($34,000) microloan...